Distilled Sensing: Selective Sampling for Sparse Signal Recovery

Distilled Sensing: Selective Sampling for Sparse Signal Recovery

8 Pages · 2009 · 236 KB · English

For each method we computed the empirical. NDP for a range of FDR levels. The curves in Fig. 2 show empirical NDP vs. empirical FDP for 10 trials of each procedure (solid lines for non-adaptive sampling, dashed lines for DS with ∆ = 0.9 and k = 5). For the same FDP, DS yields lower NDPs than non-

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whereN(¹ i;1) denotes the normal distribution with mean¹ iand unit variance The signal¹= (¹ 1; : : : ; ¹ n) is sparse if most of the components¹ iare zero Identifying the locations of the nonzero compo nents based on the dataX= (X 1; : : : ; X n) whennis very large is a fundamental problem arising in many applications, including fMRI (Genovese et al, 2002), microarray analysis (Pawitan et al, 2005), and astro nomical surveying (Hopkins et al, 2002) A common approach in these problems entails coordinatewise thresholding of the observed dataXat a given level, identifying the locations whose corresponding obser vation exceeds the threshold as signal components Suppose that the number of nonzero components of¹ grows sublinearly innaccording ton1¡¯ for¯2(0;1), and that each nonzero component takes the same (positive) valuep Suppose that instead of a single observation of a sparse signal in noise, one were able to take multiple `looks,' possibly adjusting the focus in a sequential fashion Similar adaptive methods have been proposed in the signal processing literature (Rangara jan, 2007), and they certainly are conceivable in applications such as {z } repeatedmtimes; The paper is organized as follows In Section 2 we re view the conventional nonadaptive approach to sparse recovery We introduce our adaptive sensing tech nique (DS) in Section 3, and in Section 4 we state our main results, that DS enables the recoverability of sig ni¯cantly weaker signals than standard, nonadaptive methods Section 5 provides numerical simulations of DS, and a short discussion appears in Section 6 Proofs of the main results are given in the Appendix 2 SPARSE RECOVERY BY NONADAPTIVE SENSING Consider sparse signals havingn1¡¯ signal components each of amplitudep to estimate the locations of the signal components It follows from techniques used in (Abramovich et al, 2006; Benjamini and Hochberg, 1995; Donoho and Jin, 2006; Donoho and Jin, 2008; Jin, 2003) that ifr > ¯, the procedure (2), with a threshold¿that may depend onr,¯, andn, drives both the FDP and NDP to zero with probability one asn! 1 Conversely, ifr < ¯, then no such coordinatewise thresholding procedure can drive the FDP and NDP to zero simultaneously with probability tending to one asn! 1 In other words, for the speci¯ed signal parametrization and ob servation model, the (¯ ; r) parameter plane is parti tioned into two disjoint regions In the regionr > ¯, sparse signal components can be reliably located us ing a coordinatewise thresholding procedure In the complementary region wherer < ¯, no coordinate wise thresholding procedure is reliable in the sense of controlling both the FDP and NDP This establishes a sharp boundary in the parameter space,r=¯, for largesample consistent recovery of sparse signals 3 DISTILLED SENSING fori= 1;2; : : : ; nandj= 1;2; : : : ; k, where eachÁ(j) i is nonnegative, andZ(j) iiid » N(0;1) In addition, we impose the restrictionP i;jÁ(j) i·n, limiting the to tal amount of sensing energy Note that the standard observation model (1) takes the form (3) withk= 1 andÁ(1) i= 1 fori= 1; : : : ; n Another possibility is to make multiple iid observations, but each with only a fraction of the total sensing energy budget For ex ample, setÁ(j) i= 1=p Number of observation stepsk; Energy allocation strategy:E(j) ,P k j=1E(j) ·n; forj= 1tokdo X(j) i= (q Output: Final index setI(k) ; Distilled observationsX(k) DS:=fX(k) i:i2I(k) g; Therefore, we are interested here in adaptive, sequen tial designs offÁ(j) ig i;jthat tend to focus on the signal components of¹ In other words, we allowÁ(j) ito de pend explicitly on the pastfÁ(`) i; X(`) ig i;`

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